Dependence Relations in General Relativity

Antonio Vassallo

LOGOS-BIAPUniversity of Barcelona, Department of Philosophy

Carrer de Montalegre, 608001 Barcelona

antonio.vassallo1977@gmail.com

Abstract

The paper discusses from a metaphysical standpoint the nature of thedependence relation underpinning the talk of mutual action between mate-rial and spatiotemporal structures in general relativity. It is shown that thestandard analyses of dependence in terms of causation or grounding are ill-suited for the general relativistic context. Instead, a non-standard analyticalframework in terms of structural equation modeling is exploited, which leadsto the conclusion that the kind of dependence encoded in the Einstein fieldequations is a novel one.

Keywords: Dependence relation, grounding, causation, laws of nature,general relativity, spacetime, geodesic motion, structural equation modeling.

1 IntroductionIn their comprehensive textbook on general relativity, Misner, Thorne, and Wheelerencapsulated the spirit of the theory in the famous motto “Space acts on matter,telling it how to move. In turn, matter reacts back on space, telling it how to curve”(Misner et al., 1973, p. 5). This metaphor does indeed a good job in conveying the

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basic idea behind the Einstein field equations, which can be schematically writtenas (Greek indexes run from 0 to 3):

Gµν[gµν] = κTµν[Φ, gµν]. (1)

(1) basically describes the non-linear coupling (through an appropriate constant κ)of the metrical field gµν (on which Einstein tensor Gµν depends) with the materialfield(s) Φ, whose physical properties are encoded in the stress-energy tensor Tµν.1

An interesting question to be asked then is: how much should we read intoMisner, Thorne, and Wheeler’s description of (1)? In particular, how seriouslyshould we take the reference to action and reaction? In everyday thinking, weare accustomed to the idea that matter acts on other matter in a straightforwardsense. For example, we have no problem to accept that, under certain conditions,we can act on a door in a way that determines its opening. Thus, whether theopening of the door obtains depends on whether it is pushed. However, thingsbecome much less clear and intuitive when we try to apply the same picture tospatiotemporal structures determining the geodesic motions of material bodies,or to massive bodies determining the geometry of spacetime. Even adopting askeptical attitude towards the existence of spacetime does not ease the puzzlement,because then we have to explain why Misner, Thorne, and Wheeler’s metaphor isin fact so powerful.

In the following, I will analyze from a metaphysical standpoint the determina-tion relation underlying the talk of mutual action between material and spatiotem-poral structures in general relativity. I will pay particular attention to the first partof the afore-mentioned metaphor, discussing in what sense spacetime determinesthe geodesic motion of freely-falling bodies. In the next two sections, I will reviewthe current philosophical debate on the subject, highlighting the controversy sur-rounding the notion of action. In section 4 I will introduce a non-standard frame-work for the analysis of dependence relations, which involves structural equationsmodeling. Finally, in section 5, I will show how such a framework is able toclarify the source of the controversy, concluding that the relation encoded in (1)represents a novel kind of dependence.

2 Are spatiotemporal properties causal?The most straightforward reaction one can have towards Misner, Thorne, andWheeler’s motto is to take it at face value: spacetime does not metaphoricallyguide matter, it literally causes bodies to freely fall. This plain causal reading ofspatiotemporal properties can come in different flavors. One of the most devel-oped proposals in this sense is due to Alexander Bird (see, e.g., Bird, 2009). In

1Note how the stress-energy tensor itself functionally depends on the metric tensor gµν.

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a nutshell, Bird claims that spatiotemporal properties –if fundamental– are essen-tially dispositional. Such a stance relies on the later Einstein’s invocation of anaction-reaction principle in order to account for the reciprocity encoded in the fieldequations of general relativity (see Brown and Lehmkuhl, 2016, for a discussionof the evolution of Einstein’s views on the subject). In Bird’s words:

In dispositional essentialist terms, we can see that by being potentialmanifestations of dispositional essences, spatial and temporal proper-ties may also have dispositional essences themselves. [T]hat perspec-tive is precisely that endorsed by General Relativity. Each spacetimepoint is characterized by its dynamical properties, i.e. its dispositionto affect the kinetic properties of an object at that point, captured inthe gravitational field tensor at that point. The mass of each object isits disposition to change the curvature of spacetime, that is to changethe dynamical properties of each spacetime point. Hence all the rel-evant explanatory properties in this set-up may be characterized dis-positionally.(Bird, 2009, p. 240)

In other words, Bird argues that, according to general relativity, spacetime isrecipient of change caused by matter but, given the reciprocity encoded in the Ein-stein field equations, that means that spacetime is able to cause a change in mat-ter. Under this reading, spatiotemporal properties are active in a straightforwardcausal sense, and the kind of dependence relation among material and spatiotem-poral facts underlying the Einstein field equations looks like a close relative ofeveryday causation.2 Or does it?

The main conceptual problem with Bird’s view is that possessing a disposi-tional essence necessarily entails an appropriate counterfactual.3 In the case athand, such a counterfactual would be something like “were the metrical structureto undergo a variation, the mass-energy distribution would change as well”. Theway we would evaluate the truth value of this counterfactual would be: take asolution of the Einstein field equations that depicts the actual situation 〈gµν,Tµν〉,feed a small perturbation of the metric tensor in the field equations, and if theygive back a model with a (slightly) different Tµν then the counterfactual is true.4

The talk of “small perturbations” is the general relativistic way to refer to thepossible world nearest to the actual one. The problem with this way of proceeding

2Here we are just taking for granted that dispositions are causally relevant, cf. McKitrick(2005) for discussion.

3For the time being, let us just focus on Lewis-style counterfactuals. We will later consideralternative approaches to counterfactual reasoning.

4Of course, given the reciprocity encoded in (1), the same reasoning can be repeated mutatismutandis for the dependence of spacetime geometry on the mass-energy distribution.

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is that, in general, it is not possible to non-arbitrarily fix the nearest possible world.To have a rough idea,5 imagine we have a world with just one massive body.Can we evaluate the counterfactual “were the body to disappear, the geometryof the universe would be everywhere flat”? Intuitively, we would say that thiscounterfactual is true but, in practice, we have to compare the starting model witha solution of vacuum field equations Gµν[gµν] = 0. For sure, Minkowski spacetimeis one of these solutions, but not the only one. We therefore need some additionalextra-dynamical argument to fix the Minkowski solution as the nearest world tothe starting one.

It does not take much to realize that the issues with evaluating counterfactualchanges in general relativity are a direct consequence of the dynamical nature ofthe spacetime structures described by this theory. To see this, just make a compar-ison with Newtonian mechanics. In this latter theory, the main features of space-time are fixed both in a physical –i.e. spacetime structures are not influenced bymaterial ones– and a metaphysical sense –i.e. all possible worlds in which New-tonian mechanics holds feature the same spacetime structures (see Vassallo, 2016,for an appraisal of background spatiotemporal structures from a metaphysical per-spective). As a consequence of this metaphysical rigidity, it is always possible touse Newtonian background structures as a standard against which any counterfac-tual change can be evaluated. Curiel (2015) makes a nice case:

[O]ne may be interested in the question: what would happen to theorbits of the planets in the Solar System if the sun were to vanish?Nothing simpler. Plug in [the initial-value formulation] the new val-ues for the sun’s size and mass (viz., zero), compute the new orbits,and compare them to the original ones by using the background si-multaneity and affine structures as referential framework. [...] Thereis no need for the ad hoc fixing of comparison classes of systems, orfor the ad hoc fixing of methods for identifying “the same quantityof the same system at the same place and same time, under otherwisedifferent conditions or in otherwise different states”. It’s all fixed nat-urally and canonically from the start.(ibid., p. 4)

Clearly, such a referential framework is not available in general relativity, thusmaking the evaluation of counterfactual change a rather tricky issue. The immedi-ate consequence of the troubles with Lewisian counterfactuals in general relativityis that the Einstein field equations by themselves are unable to provide a robustenough “causal-like” link between spacetime and matter.

We may ask whether there can be a different approach according to whichspatiotemporal properties can be given a straightforward causal reading without

5See Curiel (2015, section 3) for a formally rigorous example.

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resorting to Lewis-style counterfactual analysis. In other words, can we providea sound story according to which spacetime points (or small spacetime regions)6

perform an action on material bodies, which “shapes” their trajectories? If dis-positions are not enough to do the job, perhaps we can base our characterizationof action on something more physically tangible, such as some sort of energy-momentum transfer. After all, this seems to be the exact idea underlying, e.g.,the slingshot effect exploited in astronautics. Roughly, we speak of the slingshoteffect when a spacecraft passing near, say, a massive planet, gains some energy-momentum from the planet’s gravitational field and, thus, accelerates.

This well-known physical effect might seem to perfectly fit an account of cau-sation in terms of physical processes –such as the conserved quantity approach(see, e.g., Dowe, 2000). According to this view, spacetime would “push” ma-terial bodies into geodesic motion by transmitting a certain amount of energy-momentum to them. This would be a picture even more physically-laden thanthe one given in terms of dispositions. However, there are at least two reasonsto regard this position as awkward if applied to geodesic motions in general rel-ativity. The first, and most obvious, is that geodesic motions in general relativitycan be regarded as “default” motions in a sense similar to that of inertial mo-tions in Newtonian mechanics, i.e. the type of motions associated to isolated (i.e.non-interacting) bodies. Hence, it is not clear at all how the notion of energy trans-fer can be consistently introduced in the description of geodesic motions withoutrunning into troubles. For example, would that mean that there is a “true” defaultmotion that involves no energy transfer at all, which just happens to be hiddenfrom view because of spacetime’s “push”?

Secondly, general relativity teaches us to be very careful when considering lo-cal energy-momentum exchange processes. In a nutshell, the problem is that, ingeneral relativity, the gravitational energy-momentum is represented by a quan-tity –a pseudo-tensor– that is not invariant under coordinate transformations. Thismeans that gravitational energy-momentum cannot be defined locally in a unique–i.e., observer-independent– way, which in turn disrupts the general descriptionof energy-momentum transfer processes in a given spacetime region (for a techni-cal justification and a philosophical discussion of this problem see Curiel, 2000;Lam, 2011). Even if this does not automatically rule out the possibility to makesubstantial physical sense of gravitational energy (and transfer thereof) in generalrelativity (see Read, 2018, for a recent effort in this sense), still it is apparent thatdefending the conserved quantity approach in this context is a tremendously hardtask.

6This way of talking should not trick the reader into believing that the present discussion re-quires a substantivalist attitude. In fact, the relationalist can take the reference to spacetime pointsor regions as a verbal shortcut to address aspects of a web of spatiotemporal relations amongmaterial bodies.

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The above mentioned issues are exploited in Livanios (2008) to come up with astraightforward anti-causal argument. The premises of the argument are that (P1)spacetime does not force bodies to move geodesically (it just determines what theavailable geodesic paths are), and (P2) causal processes always involve the actionof forces (at least, in fundamental physics). Then the obvious conclusion is thatspacetime does not causally act on matter.

How can Livanios’ argument be contrasted? We have already seen why thedenial of (P1) is rather problematic, but, even taking (P1) for granted, still Livan-ios’ argument may be attacked by denying (P2) and, more precisely, by claimingthat general relativity is in fact a fundamental physical theory where the notionof cause can be divorced from that of force. To this line of reasoning, Livaniosreplies as follows:

Currently, there are three interpretations of what, according to GR, theinfluence of spacetime on matter amounts to: Some physicists andphilosophers think of GR as “geometrising” away the gravitationalforce, while others think of it as showing that all spatiotemporal phe-nomena are expressions of the gravitational field. Finally, there is athird view according to which, in the general relativistic context, thereis a conceptual identification of the gravitational field with the properspatiotemporal geometrical structure [cf. Lehmkuhl (2008)]. In nei-ther of these interpretations we have an indication that GR dissociatesthe concept of the cause from forces.(Livanios, 2017, pp. 24-25)

The last sentence above is rather questionable. While it is true that neitherof these interpretations of GR dissociate the concept of cause from forces, it isequally true that they do not associate the concepts either. All these interpreta-tions address is the issue of the “arrow of reduction” between spacetime and thegravitational field. This has nothing to do with the issue of causation per se.

Hence, if it is true that the problem with ascribing causal efficacy to spa-tiotemporal properties in general relativity just stems from tacitly requiring thatsuch properties perform the causal job by exerting a force, then it seems that herefriends of causation have a chance to redeem themselves. Indeed, they can justclaim that anchoring the causal talk to the presence of forces is too strong a re-quirement for causal dependence. This is exactly what Andreas Bartels does:

The answer to [Livanios’] objection is that, in General Relativity, met-rical (affine) structure is the means by which the gravitational fieldcouples to matter. Spacetime affects matter not by means of classi-cal forces, but by means of its metrical (and affine) structure which

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brings about tidal forces producing paradigmatic causal effects likespatial deformations of material bodies.(Bartels, 2013, p. 2005)

Bartels bases his causal thesis on a counterfactual analysis of causation alter-native to the standard Lewisian analysis considered so far, namely, the interven-tionist (or, manipulationist) framework (see Woodward, 2016, for an introductionto the subject).

The core idea behind the interventionist approach is that a condition C causesa condition E if and only if were C to be subjected to an exogenous process thatchanges it, E would change as well. More prosaically: C causes E if and onlyif by wiggling C, E wiggles as a result. Of course, the intervention behind thewiggling of C does not have to be agential: any physical process external to thesystem under scrutiny would do (hence the word exogenous used above). Insistingthat an intervention has to be external to the system is very important, because wewant to exclude that the wiggling of E is a consequence of the change in anothercausal link that bypasses C. That being said, it is not necessary to claim that anintervention has to be a possible physical process in a strict sense.7 All we requireis (i) to have a “coherent conception” of what it is to wiggle C, and (ii) to be ableto exclude “confounding” cases in which the change of E is not (just) due to thewiggling of C.8 Hence, according to Woodward:

This suggests that there will be a basis for claims about what willhappen to E under an intervention on C as long as we can associatesome well-defined notion of change with C and as long as we havesome grounds for saying what the effect, if any, on E would be ofchanging just C and nothing else.(Woodward, 2003, p. 131)

In the case of general relativity, according to Bartels, this basis clearly exists,and is represented by the formal machinery of the theory. Indeed, if we modelboth C and E via some variables related by the laws of the theory, we are able toevaluate any kind of counterfactual claim whose antecedent involves a change of

7To be fair, this is a contentious point. Some may object that, given that a cause is basicallya “handle” to manipulate effects, what it is to be a cause is defined by the very nature of theassociated intervention. Hence, it is quite suspicious to allow for interventions that are not realiz-able –at least in principle– through a physically well-defined process. We will return to this pointin section 4. For the time being, we just stress the fact that there can be manipulationist accountsstronger than the one discussed here, in the sense that they place more strict constraints on allowedinterventions.

8This further clarifies the sense in which an intervention on C has to be external to the system:it has to sever any confounding causal connection “upstream” of C (in particular it must be thecase that C cannot affect E via any other causal route than the one probed by the intervention).

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value in the variable modeling the condition C, because it is the laws of the theorythemselves that provide a coherent conception of change and a non-confoundinglink between antecedent and consequent. Let’s see how this works in the concretecase of geodesic motions.

First of all, we model material motions by timelike curves xµ(λ) (λ represent-ing an appropriate affine parameter): these are our dependent variables. The dy-namically allowed curves –more precisely, the curves representing freely-fallingbodies– are the solutions to the equations of motion of general relativity9 (a dotindicates standard differentiation with respect to λ):

xµ + Γµνσ xν xσ = 0. (2)

Equations (2) follow directly from (1) (see, e.g., Misner et al., 1973, section 20.6,for a formal derivation) and represent the mathematical codification of the factthat freely-falling (i.e. non-interacting) bodies move along the “straight lines”of the geometry encoded in gµν. The measure of the “straightness” of a curve isprovided by a mathematical device called connection (usually symbolized by thecovariant derivative operator ∇), whose components are the so-called Christoffelsymbols Γ

µνσ appearing in (2). These Christoffel symbols are derived from the

metric tensor through the relation

Γµνσ =12

gρµ(gσρ,ν + gρν,σ − gνσ,ρ

), 10 (3)

where the comma indicates standard differentiation with respect to the subsequentindex.

With this machinery in place, we proceed by indicating the exogenous vari-able, i.e. the one to be wiggled. In our case, this is clearly the metric tensor gµν.Let’s now consider a certain solution of the field equations 〈gµν,Tµν〉 (plus relatedboundary conditions, if any). Such a model will have a set of geodesic motions{xµi (λ)}i∈N given by the geodesic equation. The wiggling of gµν here amounts toselecting a different metric gµν which is dynamically allowed, i.e., such that it ispart of another model 〈gµν, Tµν〉. We then feed gµν into the formula (3) for thecoefficients of the connection and, consequently, in the geodesic equation. Ofcourse, the new set of solutions {xµi (λ)}i∈N of (2) will be different from the startingone {xµi (λ)}i∈N. Hence, by intervening –in the loose sense justified above– on thegeometry of spacetime, we changed the geodesic motions of material bodies. Thecausal verdict is thus positive: the geometrical properties of spacetime do causebodies to move with geodesic motion. Note how it is the theory itself that pro-vides the conceptual apparatus needed to make clear (i) what it means to change

9See Tamir (2012) for a voice against this “canonical” view of geodesic motion.10More precisely, this relation holds for a very particular kind of connection, called Levi-Civita.

I will come back to this point in section 5.

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the cause C, and (ii) how the depending condition E would change as a direct andexclusive consequence of the wiggling C.

By basing his causal analysis on interventionism, Bartels is able to defusethe objection that a causation-friendly treatment of spacetime in general relativityhas to imply or require that spacetime literally pushes material bodies to movegeodesically. In fact, the manipulationist framework does not seek to reduce thenotion of cause to any more fundamental notion (let alone that of force, or energy-momentum exchange). Of course, the fact that the interventionist account of cau-sation is non-reductive is not a problem in the context of our discussion, sincewe are interested in assessing whether there is a causal link, and not in giving adetailed conceptual account of what causation is.

How does Bartels’ proposal fare with respect to the issues with counterfac-tuals evaluation in general relativity? Unfortunately for friends of causation, theanswer is: not that well. In fact, the manipulationist analysis sketched abovedoes not work on “small portions” of a model, that is, we cannot define surgicalinterventions that affect only a condition local C while leaving everything elseuntouched (which, again, is a key requirement if we want to avoid confoundingcases).

In the above example, the intervention works insofar as the geometry –thatis, a diffeomorphic class of metric tensors– over the whole model is changed:it makes no sense to ask what would have happened if we changed it just in aneighborhood of a point, leaving everything else unchanged. To see why it is so,let’s assume that the model is well-behaved enough to admit a 3+1 decompositionof the dynamics. What we would do to represent this local intervention wouldbe to take the values of the appropriate variables on an initial Cauchy 3-surface,make a “small deformation” (that is, changing the value of the metric variables)on a small neighborhood of the surface, and let the dynamics evolve the surfaceto subsequent ones.

The problem is that, in this formulation, the Einstein field equations repre-sent a geometrical constraint over the sequence of 3-surfaces embedded in a 4-dimensional model. Therefore, deforming –even infinitesimally!– the geometryof the initial 3-surface would most likely violate the embedding constraints and,hence, the field equations themselves. Jaramillo and Lam (2018) give a detailedtechnical justification for why it is so. Very roughly speaking, the geometricalconstraints can be cast as a set of non-linear and elliptic differential equations.The non-linearity implies not only that a combination of solutions of the systemof equations does not constitute a solution by itself, but also that adding a “smalldeformation” to a solution will not give in general a new solution. The elliptic-ity of the system, on the other hand, points at the fact that the equations do notstrictly speaking describe any dynamical evolution for the initial data but, rather,they encode global constraints for such data: this implies that the value taken on

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by a solution at a point generally depends on all the initial data over a Cauchy sur-face. This is why it is extremely difficult to define a surgical intervention without“messing up” the whole model.

To sum up, if an intervention is a dynamically allowed change in the value ofa variable, then interventions can mostly be “model-wide”, and causation losesany interesting local connotation, i.e. setting aside few extremely favorable cases,we cannot counterfactually test any causal link that would involve a surgical inter-vention on a small part of the model (this problem is also discussed in the contextof frame-dragging effects in Hoefer, 2014, section 3). This is awkward not onlyfrom an ontological perspective, but also from an explanatory one. For example,we would be forced to say that the precession of Mercury’s perihelion is not aneffect just due to the surrounding spacetime geometry, but to the whole spacetimegeometry of the solar system.

In short, buying into this type of “causal non-locality” would push towardsa very strong type of holistic metaphysics, and would also imply that any expla-nation of local state of affairs that mentions only the immediate spatiotemporalsurroundings of such states of affairs would be incomplete. This is quite a highprice to be paid to ascribe a causal nature to spatiotemporal properties. So, allthings considered, switching to an interventionist counterfactual approach is notthe panacea for defenders of the causal nature of spatiotemporal and material in-fluences in general.

To conclude our review of the literature discussing the causal nature of spa-tiotemporal properties in general relativity, let us consider an anti-causal argumentput forward in Katzav (2013). The gist of the argument is that being committedto a causally-friendly view of spatiotemporal properties entails a commitment tophysically opaque instances of causation.

Katzav starts by noticing that a cause can bring about a non-occurrence in aphysically acceptable sense only as a result of the inhibition of some endeavor.For example, imagine that a skinny hiker encounters a huge boulder on his pathand unsuccessfully tries to push it out of his way. In this case, we may say thatthe non-occurrence of any boulder’s movement is caused by its huge mass. Aphysicist would nicely describe such a scenario in terms of Newton’s second lawas an interaction between the hiker and the boulder, in which the boulder pushesthe ground “harder” than the hiker pushes the boulder. Thus, here, the mass ofthe rock does an active job in preventing something –i.e. its sudden movement–from happening. However, it would be physically awkward to claim that the hugemass of the boulder causes the non-occurrence of its sudden acceleration whenno external force is exerted on it. In this second case, it does not seem right tosay that the rock’s mass does the job of preventing it from accelerating. In fact, aphysicist would explain the absence of this kind of motion by appealing to a non-causal principle such as Newton’s first law. Insisting that, also in this case, mass

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plays an active causal role would clearly entail a commitment to spooky instancesof causation.

The next step in Katzav’s argument is to translate this line of reasoning togeodesic motions in general relativity. Thus, the question is: can we say that thegeometrical properties of spacetime cause free bodies to move geodesically? Hav-ing in mind Katzav’s criterion for physically acceptable instances of causation, wemight be tempted to say that, yes, spacetime causes geodesic motions to happenby countering the bodies’ disposition to move non-geodesically.

However, such a picture is untenable for two reasons. First, similarly to theenergy-transfer case discussed earlier, it would seem that now the “natural” stateof motion of a body is opposed by spatiotemporal structures, and geodesic motionsare the result of such an opposing influence. This is, of course, not the storytold by general relativity, according to which geodesic motion is an instance ofpersistence as is through motion –i.e. absence of change in any relevant physicalrespect throughout the trajectory. Secondly, the explanation of geodesic motionsin this picture would look like the enactment of a cosmic conspiracy according towhich the true nature of free motions is hidden from our view.

If, instead, we admit that geodesic motions are not an effect of counteringhidden true motions, then the causal efficacy of spatiotemporal properties is calledinto question as unfit to back up the physical explanation of the phenomenon.Indeed, the most we can say in this case is that spacetime causes the absence ofinexplicable changes in the trajectories of freely-falling bodies. For sure, this isnot a “physics-friendly” instance of causation.

Katzav’s anti-causal argument is probably the most contentious among theones reviewed in this section. What is most puzzling about Katzav’s argumentis the premise that a cause can bring about a non-occurrence in a physically ac-ceptable sense only as a result of the inhibition of some endeavor. Katzav hereis clearly brushing aside alternative –and equally acceptable– instances. For ex-ample, spacetime might cause the non-occurrence of a certain trajectory xµ(λ) bybringing about another trajectory xµ(λ) that is incompatible with xµ(λ).11 Giventhat much of Katzav’s argument depends on this premise, it looks like he is provid-ing the least cogent anti-causal case of the batch presented in this section. How-ever, this is just a cold comfort for the fans of causation in general relativity, giventhe much more troublesome issues discussed earlier –in particular, the problem ofmeaningfully define counterfactual change in Lewis-style analysis, and the issuewith non-local interventions in the manipulationist approach.

11I am grateful to an anonymous referee for suggesting this objection.

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3 Are spatiotemporal properties non-causal?So far, we have discussed in detail why and how general relativity does not rep-resent a hospitable environment for fans of causation. Consequently, skeptics to-wards causation seem to gain the upper hand in the debate. Or do they? Otherwisesaid, can causal skeptics come up with a convincing metaphysical characterizationof the spacetime/matter-dependence?

Let’s go back to Livanios’ work, and see his own proposal:

[A] more cautious analysis points out another interpretation, accord-ing to which the dynamical character of space-time allows the trans-world variation of space-time structure in such a way that the Ein-stein equations hold. In such an approach, we look upon the Einsteinequations as giving a law-like consistency constraint upon the jointfeatures (space-time structure and mass-energy distribution) of any(physically) possible world.(Livanios, 2008, p. 389)

Judging from the above quotation, it seems that Livanios has in mind somesort of “physical possibility” relation, e.g., the geometry of a certain spacetimeregion A determines (through the mediating role of (1)) the “physicality” of a cer-tain trajectory xµ(λ) in A. We immediately see the problem with this proposal: itassigns an inherently modal nature to the dependence relation underpinning theEinstein field equations, and we already know that modality in general relativityis a tricky issue. Indeed, such a relation (again, through the mediating role of (1)),should support counterfactuals like “if the geometry of A were different, it wouldbe physically impossible for a freely falling object to move along xµ(λ) in A”. Inevaluating such a counterfactual, we would face the very same troubles alreadyencountered before. As already pointed out in section 2, because of the dynamicalcharacter of spacetime, there is no objective trans-world standard against whichwe can assess any notion of counterfactual variation. So there is no straightfor-ward way to individuate the “changed version” of A in a possible world near tothe starting one (to catch a glimpse of the debate on trans-world individuation ofspacetime points and regions, see Earman, 1989, chapter 9, section 14).

The immediate reaction to my criticism is that it is unwarranted and stemsfrom a misguided reading of Livanios’ passage. In fact, Livanios’ quotationshould be intended as stating that this physical possibility relation is just a “model-building” relation that couples metric tensors with stress-energy tensors in accor-dance with (1). If that is the case, then all that we need is that only worlds inwhich the Einstein field equations hold are physically possible. In particular, wedo not need to say anything about the evaluation of counterfactual change, possi-ble worlds vicinity, or anything like that.

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Let’s accept for the sake of the argument that this is the case. Then a newsubstantial issue presents itself: Livanios’ possibility relation becomes trivial tothe point of mere analyticity, thus morphing into some sort of conceptual neces-sity relation. The main trouble with the relation of conceptual necessity is thatit usually supports very feeble explanations. For example, once we define whata bachelor is, it becomes matter of conceptual necessity that “X is a bachelor”obtains whenever “X is an unmarried man” obtains. However, the analytical linkbetween these two facts makes an explanation of the form “X is a bachelor becauseX is an unmarried man” rather uninformative. In a similar vein, an explanation ofthe form “such and such geometry is coupled to such and such mass-energy dis-tribution because the Einstein field equations hold” is not very informative. Themost that we can learn from such an explanation is that the explanandum is notmerely an accident, in the sense that it is a particular instance of the law-like gen-eralization constituting the explanans. This might be seen as some relevant pieceof information, but the fact remains that a scientifically satisfactory explanationshould mention much more then just the mere fact that (1) hold –for example, if(1) are cast in Hamiltonian form in a way that entails a 3 + 1 decomposition of thedynamics, then some boundary conditions together with the specification of theinitial conditions on a spacelike Cauchy surface would surely enter a satisfactoryexplanation of the dynamical development.

Hence, it is doubtful whether Livanios’ relation of conceptual/nomic necessitycan be enough to do any interesting explanatory job. If that is all that the anti-causal party can say about dependencies in general relativity, then they do notseem to fare any better than the friends of causation in the debate.

At this point, if we want to make some progress, we need to pinpoint someminimal desiderata that a convincing non-causal characterization of the depen-dence relation underlying (1) should fulfill, and then see if metaphysicians havesomething interesting to propose along these lines.

The first two desiderata should be basically a non-causal counterpart of theprominent features of causal dependence. Hence, the first requirement is thatsuch a relation has to have explanatory power: for example, it should providea conceptually robust backbone for explaining inertial motions via geometricalstructure. The second desideratum is that it has to be a relation of determinationin a strong ontological sense, that is, the existence of a condition determiningthe existence of another (e.g., the fact that freely-falling bodies move in suchand such a way obtains in virtue of the fact that spacetime has such and suchgeometry). However, to avoid the problem that causal relations face in generalrelativity, we should further require this relation not to be inherently modal, thatis, not to require modal notions in order to be defined (so supervenience wouldnot do for our purposes). Finally, in order to avoid triviality, we should excludethat such a relation is just conceptual necessity.

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After a moment of reflection, we realize that there might be a good fit for thisdescription. Indeed, the above desiderata are fulfilled by a dependence relationthat metaphysicians call metaphysical grounding. The notion of grounding is alively debated topic among metaphysicians and, for obvious reasons, we cannotgo through the debate here (see the essays in Correia and Schnieder, 2012 to havean idea of the state-of-the-art on the subject). For this reason, we will pick aparticular view on grounding, nicely exemplified by Paul Audi:

Grounding is the relation expressed by certain uses of the phrase “invirtue of ,” as in “the act is wrong in virtue of its non-moral proper-ties.” [...] We should not use “in virtue of” where it might expressa reflexive relation, such as identity. Since grounding is a relation ofdetermination, and closely linked to the concept of explanation, it isirreflexive and asymmetric. [...] On my view, grounding is a singu-lar relation between facts, understood as things having properties andstanding in relations. Facts, on this conception, are not true proposi-tions, but obtaining states of affairs.(Audi, 2012, pp. 102-103)

This brief characterization of grounding might make sense in general relativ-ity: the Einstein field equations describe how a set of material (or spatiotempo-ral) facts obtain in virtue of a set of spatiotemporal (or material) facts obtaining.The word “obtaining” here has to be understood as ontological rather than simplyconceptual determination. In this vein, grounding can be thought of as a partialordering with respect of fundamentality, while causality is a partial ordering withrespect to time. This feature sounds good, because it means that grounding isbetter-suited than causation for ordering facts about time, which are encoded in a4-dimensional geometry.

So, have we finally selected a good candidate for the role of dependence rela-tion is general relativity? Of course not!

There are at least two problems with metaphysical grounding as the relationunderpinning the dependence described by (1). The first is that, on this view,Misner, Thorne, and Wheeler’s motto becomes at best an awkward metaphor.The problem is not stylistic, but substantial: nobody would contend that thereis something straightforwardly physical in the way spacetime and matter act oneach other, and the notion of metaphysical grounding does not seem to capturesuch a physical character in any way. For sure, nobody would claim that the sling-shot effect clearly exemplifies a case of metaphysical grounding without feeling abit embarrassed. The second –and even worse– problem is that, according to (1),material facts depend on spatiotemporal ones and viceversa. Hence, the require-ment of asymmetry for grounding is in stark tension with the (prima facie) mutualdependence encoded in the Einstein field equations. Otherwise said, the laws of

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general relativity do not favor any claim regarding geometrical facts being morefundamental than material ones or the other way round. One may try to resistthis conclusion by pointing out that the stress-energy tensor in (1) depends on themetric tensor, thus claiming that it is geometric facts that ground material facts ina clear formal sense, and not viceversa. However, that would be too hasty: whileit is true that the stress-energy tensor depends on the metric tensor, we shouldnot overlook the fact that it also depends on material fields. Hence, the most wecan say is that the stress-energy tensor represents some sort of relational propertyshared by geometry and material fields, but this does not make any of the twomore fundamental than the other (cf. Lehmkuhl, 2011 for a nice historical andphilosophical discussion of this point).

We have finally reached the core of the issue at stake. According to the po-sitions reviewed in this section and the previous one, it is quite uncontroversialthat the Einstein field equations encompass a dependence relation between ma-terial and spatiotemporal facts. However, nobody so far has been able to comeup with a convincing characterization of the dependence involved. For sure, sucha relation is explanatory in nature, it is not identity (unless (1) are taken to de-fine a stress-energy tensor, which is controversial to say the least), and it is notinherently modal (although it might have modal consequences, depending on theparticular view of modality adopted). However, it is too “physics-friendly” to bejust grounding yet not “physically active” enough to be just causation.

My take on the problem is that general relativity is trying to teach us a philo-sophical lesson here, namely, that we are drawing too sharp a line between ground-ing and causation. In the next section I am going to make this intuition clearer byhighlighting the well-known structural similarities between grounding and cau-sation. In particular, I will draw on a recent proposal to analyze both groundingand causation via a generalized formal framework designed to “spot” dependencerelations among facts.

4 A unified analysis of grounding and causationAs already said, the conceptual similarities between grounding and standard cau-sation have not gone unnoticed in the philosophical literature. To begin with, theyare both usually modeled as irreflexive, transitive and, hence, asymmetric binaryrelations i.e. strict partial orderings. Moreover, it can be easily argued that theyboth take facts as relata. In a nutshell, talk of causation between events can bereduced to talk of causation between facts about events happening (see Mellor,1995, for an extensive articulation and defense of this view). If we accept thissimilarity between the two, then it is easy to extend this analogy to the distinctionbetween partial/full ground on the one hand, and contributory/sufficient cause on

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the other. But the structural similarities between grounding and causation rundeeper than this. This fact becomes apparent if we analyze textbook cases of bothof them through the structural equation modeling (SEM) framework. SEM is anumbrella term that groups together many mathematical and statistical tools usedto investigate possible ordered structures underlying huge datasets (see Westland,2015, for a historical survey of this rather heterogeneous set of methodologies).For our purposes, we are interested in the particular framework for testing causallinks originally developed in Pearl (2000). As it will become apparent in a mo-ment, such a framework shares some characteristic traits with the “generic” inter-ventionist approach advocated by Bartels and discussed in section 2. However, theSEM framework shows some new interesting features that makes it more suitablefor being applied in the context of general relativity.

Without going too much into technical details, a structural equation model forcausation consists in a set of variables that can be further grouped into endoge-nous (i.e. those variables modeling the dependent conditions in the causal chain),and exogenous (those variables that bring about the effects in the causal chain).Exogenous and endogenous variables are related by a set of equations that givea quantitative description of the dependence relations involved. These equationsare basically functional relations, where the independent variables are exogenousand the dependent variables are endogenous. There is no a priori constraint on thetype of functions involved, the only desiderata being that (i) the model has to ade-quately reproduce the correlation patterns observed in the original dataset, and (ii)the causal chain has to be consistent with the general principles used to justify themodel. It is important to note that, since the framework is able to describe struc-tured dependence chains, the exogenous/endogenous distinction is relative: givena certain link of the chain, all variables “downstream” will be endogenous and allthose “upstream” will be exogenous. To fix the ideas and keep things simple, inthe following we will mainly discuss toy cases of single-link chains, so that the ex-ogenous/endogenous distinction becomes absolute. Once the way the frameworkworks becomes clear, generalization to more complex chains is straightforward.

How can we say that a model adequately reproduces the correlations? Verysimply, we just assign to the exogenous variables the actual values of the datasetand we see if the functions posited give back the actual values of the endoge-nous variables. Once this is achieved, we test the consistency of the causal chainby wiggling each of exogenous variables (i.e. assigning to them values differentfrom the actual ones) and see if the functional dependence determines an appro-priate wiggling of the endogenous ones. This procedure becomes clearer if werephrase it in terms of manipulationist counterfactuals: “if an intervention wereto set an exogenous variable to such and such value, the endogenous variables

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would have taken on such and such values”.12 If the counterfactual pattern gen-erated in this way is consistent with the particular laws underlying the processunder scrutiny, then the model can be considered as successfully catching a causalstructure underpinning the correlations observed in the dataset.

The standard toy example that shows how the SEM framework “spots” causa-tion involves the correlation between the shattering of a window on the throwingof a stone. In this case, if we take the laws of classical mechanics to be true, wewould guess that the dependent part of the system is the window. We thus usean endogenous variable “S ” to model the fact that the window shatters and we letthis variable take on two values {0, 1} depending on whether this fact obtains ornot. Likewise, we employ the exogenous variable T ∈ {0, 1} to model whether thefact that the stone is thrown obtains or not. The functional dependence in this casewould be simply S ←

= T . Notice that the symbol “←=” highlights that the relationbetween the two variables is asymmetric.

Imagine that our dataset concerns one hundred repetitions of an experimentconsisting in throwing a stone of fixed mass, shape, and initial velocity againsta standardly crafted window, showing that the window shatters in 99% of thecases. Then our assignment of the actual value of the exogenous variable wouldbe T = 1, which gives S = 1. So the model adequately reproduces the correlationsin the dataset.13 The next step would then be to wiggle T and see if the ensuingcounterfactual patterns are consistent with the laws of materials science. We getthat “were an intervention to prevent the stone from being thrown, the windowwould not have shattered” is true under the model (setting T = 0 leads to S = 0),while the counterfactual “were an intervention to prevent the stone from beingthrown, the window would have shattered” turns out to be false as required, to-gether with the counterfactual “were the stone to be thrown, the window would nothave shattered”. Moreover, by construction, the model makes backtracking coun-terfactuals such as “were a manipulation to shatter the window, the stone wouldhave been thrown” false, because an intervention on the window would sever thelink that S has with T (which implies that any manipulation on T does not affectS ).

The result of our analysis is that our model works, so it can reliably be takenas representing a genuine causal dependence underlying the correlations in thestarting dataset. Note how this treatment inherits all the advantages that a manip-ulationist account of causation has against the usual Lewisian analysis. First of all,

12Also in this case, in order to avoid confounding cases, it is required that an intervention on avariable disrupts all the links upstream of this variable.

13The model can be easily expanded to account for more complex situations. For example, ifthere are two stone throwers, then we have to add another exogenous variable T ′ ∈ {0, 1} andanother structural equation, i.e. S ←

= T ′. This also shows how the framework deals with cases ofcausal overdetermination.

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the framework dispenses with the talk of possible worlds. Second, it avoids back-tracking counterfactuals by construction. Third, it permits not only the analysis oftoken cases of causation (i.e. the stone with such and such characteristics thrownat the window crafted in such and such way), but also of type cases (the equationS ←

= T is totally general and applies to all cases of a single stone being thrown at awindow). Fourth, the analysis is quantitative in nature,14 which permits a deeperunderstanding of the causal dynamics involved.

That being said, the SEM framework so sketched differs substantially fromthe manipulationist approach discussed in section 2. The most striking differencebetween the two can be tracked down –recalling Woodward’s quotation on page7– to the theoretical basis for assessing “claims about what will happen to E underan intervention on C”. In the approach discussed in section 2, interventions wereevaluated directly against the laws or principles of the particular theory (or set oftheories) which constituted a viable theoretical basis for the context under consid-eration. In this way, such laws had a twofold role, namely, to provide a coherentnotion of what a manipulation amounts to, and to constitute a formal machineryfor evaluating the effects of any given manipulation. To get a more vivid idea ofhow this works, let’s return to the stone/window example. In this case, as alreadysaid, a viable theoretical framework to back up the interventionist analysis mightbe Newtonian mechanics and, in particular, the laws of inelastic scattering. Notehow these laws give a clear physical description of the situation we are analyzingand provide the formal machinery for the quantitative evaluation of concrete casesin which the stone is (or is not) thrown against the window.

In the SEM framework, on the other hand, such a formal machinery for quanti-tative evaluations is provided by the structural equations themselves. Now, whilesuch equations surely have to be consistent with the laws and principles of thetheoretical framework adopted to justify the model, they may or may not coincidewith the actual form of the laws or principles involved in justifying the adoptionof the model, depending on the level of detail required by the analysis. Again,in the stone/window example, the equation S ←

= T is not a physical descriptionof inelastic scattering, but it models how some (physical) facts would bring aboutother facts were the laws of inelastic scattering to be true: If all we want to testis the systematic obtaining (or non-obtaining) of S under the condition that Tobtains (or does not obtain), then the model is detailed enough. If, instead, wewanted a detailed analysis of, say, how far the glass debris fly in relation to theinitial momentum of the stone, then we would use a much more complicated set ofstructural equations, which would coincide with the laws of Newtonian dynamics

14This means that we can create more complex models where variables do not have to be binary.For example, we can have a more accurate stone/window model with stone’s momentum as theexogenous variable and glass’ deformation as the endogenous one.

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(or be a suitable approximation thereof).This distinction between structural equations and underlying laws has a clear

bearing on the constraints that an intervention has to obey. For, if the model co-incided with the actual dynamical description of the modeled process, any inter-vention had to be directly encoded in the fundamental equations of the underlyingtheory. However, in the SEM framework there is no such restriction, the only con-straints being that the model be logically consistent with the underlying laws orprinciples, and correctly reproduce the correlations in the dataset.

This fact makes it possible in many cases to come up with structural equationswhich feature variables whose range can include values associated with physicallyor even metaphysically impossible interventions. This liberal attitude towards thewiggling of the exogenous variables is not a problem, because interventions bythemselves are to be seen as a conceptual stress test for the structure of the model:if such a structure reproduces the observed correlations and remains consistentwith the underlying justificatory principles under any “strain”, then the model is asolid one. If you want, you can picture a causal model as a system of connectedsprings inside the rigid box of the justificatory principles invoked: it provides ex-tra conceptual degrees of freedom to an otherwise conceptually rigid apparatus.This peculiarity of the SEM analysis looks very encouraging in view of applying itto the general relativistic context. Indeed, the main problem with Bartels’ manip-ulationist proposal was that evaluating counterfactuals directly against the laws ofgeneral relativity dramatically constrained the scope of manipulations and, con-sequently, the effectiveness of the dependence analysis. In the SEM framework,counterfactuals would be evaluated against a structural model, thus giving moreroom for analytical maneuver, while still remaining consistent with the laws ofthe theory. We will elaborate more extensively on this point in the next section.

At this point, however, the objection already presented in footnote 7 becomesall the more daunting. Let’s repeat it here: the nature of a cause is deeply en-tangled with the nature of the associated manipulation; hence, physically or evenmetaphysically impossible interventions would make for conceptually obscure –ifnot ill-defined– causal conditions. Quoting again Woodward:

[T]he claim that an asteroid impact caused the extinction of the di-nosaurs can be understood within an interventionist framework as aclaim about what would have happened to the dinosaurs if an inter-vention had occurred to prevent such an asteroid impact during therelevant time period. In this case we have both (i) a reasonably clearconception of what such an intervention would involve and (ii) princi-pled ways of determining what would happen if such an interventionwere to occur. By contrast, neither (i) nor (ii) hold if we are asked toconsider hypothetical interventions that make it the case that 2+2 , 4

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or that the same object is at the same time both pure gold and purealuminum or that transform human beings into houseflies. Causalclaims that require for their explication claims about what would hap-pen under such interventions [. . . ] are thus unclear or at least have nolegitimate role in empirical inquiry.(Woodward, 2008, p. 224)

A possible reply to this objection is that the characterization of causal condi-tions on the SEM framework shifts the accent from the nature of manipulationsto the aptness of the corresponding variables in modeling the particular situationanalyzed. For example, we can minimally require that apt variables should takeon values that represent distinct conditions, that distinct values of the same vari-able should represent mutually exclusive alternatives, and that variables representenough conditions to capture the structure of the situation being analyzed (cf.Blanchard and Schaffer, 2017, section 1.3, and references therein, for a discus-sion of aptness constraints). The important point is that these aptness constraintsmake the nature of causal conditions and their change intelligible and meaningfulto the particular context considered. For example, there is no particular prob-lem in constructing a scientifically justified causal model that backs up a claimlike “If human beings suddenly turned into houseflies, urban environments wouldcollapse”, even if no scientific theory would give you a basis for characterizingthe change of a variable from “human being” to “housefly”. Likewise, we canperfectly understand that a manipulation that sets the variable to, say, “Vulcan”instead of “housefly” would lead to opposite effects on urban environments (sincewe all agree that Vulcans are more rational than both houseflies and human be-ings). Of course, the skeptic might still point out that such an approach worksonly on a case-by-case basis, especially when we already have a fairly clear ideaof where to look in order to spot a dependence. To this claim I will not replyhere, being content to notice that the analysis of spacetime/matter-dependenciesin general relativity is in fact one of these felicitous cases.

The liberal attitude towards interventions is a key assumption if we want toshow that the SEM framework analyzes cases of grounding in the same way as ithandles standard causation. Here I will mainly draw from Schaffer (2016, section2) and Wilson (2017, section 5), in which such an extension of the framework togrounding is proposed and worked out.

Consider now a textbook case of grounding, namely the existence of singletonSocrates being grounded in the existence of Socrates. Clearly, our intuition here isthat the fact that “singleton Socrates exists” obtains is dependent on the fact that“Socrates exists” obtains. Let’s call the endogenous variable modeling the formerfact S ∈ {0, 1}, and the exogenous variable modeling the latter fact T ∈ {0, 1}. Notsurprisingly, the structural equation modeling this dependence would be S ←

= T .

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The assignment of the actual value to T would be 1, which gives S = 1: thismodels the truth of “If Socrates exists, then singleton Socrates exists”. Now wewiggle T to zero and, as a consequence, S becomes zero. This symbolizes thetruth of the counterfactual “were an intervention to prevent Socrates from existing,singleton Socrates would not have existed”, which is exactly what we would haveexpected given our metaphysical treatment of grounding.

Moreover, the model falsifies by construction the counterfactual “were an in-tervention to prevent singleton Socrates from existing, Socrates would not haveexisted” (see Wilson, 2018, section 4, for a detailed analysis of how the SEMframework treats this counterfactual). Note that the antecedent of this counter-factual is metaphysically impossible. This is the case not because there are nometaphysically possible ways for singleton Socrates to fail to exist, but because,in this context, an external intervention that directly sets S to 0 would sever themetaphysically necessary link between the ground T and grounded S . Just to beclear, an intervention consisting in killing Socrates (or Socrates’ mother) wouldnot count as a direct (external) intervention on singleton Socrates, since it wouldin fact exploit the link from T to S . If the reader is now having a very hard timethinking of a way to directly intervene on S without passing through T , then sheis starting to grasp what a metaphysically impossible intervention means in thiscontext (cf. Wilson, 2017, section 6, for a discussion of this type of impossibleinterventions). This, in a nutshell, is why this generalized SEM framework needsto allow for metaphysically impossible interventions.15

If we agree on this analysis, then it is clear that the above statement depictsa counterpossible situation on the considered model. Now, according to standardcounterfactual semantics, counterpossible statements are trivially true, which isclearly unacceptable in this context. This is why this treatment demands a new,non-standard semantics to be built up (see, again, Wilson, 2018 for a preliminarydiscussion of this issue). Also in this case, the framework is able to account forboth the token case involving Socrates, and the type case involving a singletonand its element.

Once the analogy between the stone/window and the element/singleton casesis acknowledged, it is easy to see how the framework can be pushed further tohandle dependencies involving absences. For example, we can have T ∈ {0, 1}modeling the fact that I water the plant, and S ∈ {0, 1} modeling the fact thatthe plant dies. According to the laws that govern the physiology of plants, thestructural equation in this case would be S ←

= 1 − T . Readers without a greenthumb have certainly experienced the validity of this model. It is a simple exercise

15To be fair, some authors (e.g. Leuenberger, 2014) tend to deny that the obtaining of thegrounding fact always necessitates the obtaining of the grounded fact. However, in order to defendthe need for metaphysically impossible interventions in the generalized SEM framework, we justneed that necessitation between ground and grounded holds at least sometimes.

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to show that this model works perfectly well also if we say that T models the factthat unicorns exist and S models the fact that the set of unicorns is empty. Ofcourse, these analogies can be pushed even further, to embrace more complexmodels.

To sum up, if we generalize the SEM framework for causation to include a“liberal” view of manipulationist counterfactuals and a non-standard semanticsfor counterpossibles, then we obtain a powerful analytical tool to analyze not onlyinstances of causation, but also instances of grounding. This generalized SEMframework is very peculiar in that, from a structural perspective, it is blind to anydistinction between grounding and causation. For example, it does not distinguishthe stone/window case from the element/singleton case: from the framework’spoint of view they are both instances of a structural dependence of the form X ←

= Y .However, there is a clear sense in which, even in the generalized SEM framework,we can still distinguish grounding from causation. This distinction basically boilsdown to the “mediating” or “formative” principles used, that is, the set of prin-ciples invoked to justify the structural relations of a model. If these principlesare just laws of nature (as in the stone/window case), then we are dealing with aninstance of causation; if just metaphysical or mathematical principles are invoked(as in the element/singleton case), then that is a case of grounding (see Wilson,2019 for a thorough defense of this law-based demarcation criterion against moreusual criteria, such as that stating that causation is a diachronic relation, whilegrounding is not). Nevertheless, such a distinction is strictly speaking external tothe SEM framework, i.e., it is superimposed on the formalism without having anybearing on the results. From an internal point of view, what matters is just thefunctional dependencies established.

So, how far should we take the structural analogy between grounding andcausation highlighted by the generalized SEM framework presented? The mostconservative reaction is not to read too much into this. Even if grounding andcausation might share some similar features, this is not enough to make the casefor them to be related in a way that goes beyond some sort of vague conceptualresemblance (e.g. both being mentioned in explanation-related context), let aloneconsidering them as metaphysically related (e.g. one being reducible to the other).However, nothing speaks against taking the grounding/causation analogy moreseriously. In fact, there is a class of SEM models where, I believe, the idea ofgrounding and causation being somewhat metaphysically related gains traction.This class of models involves mixed chains of dependencies.

To have a down-to-earth example, imagine a case in which a football playercrosses the ball inside the penalty area and hits a defender’s hand, thus determin-ing a penalty kick. An appropriate functional model of this situation would bejustified on the basis of both the laws of physics (the ball being kicked in such and

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such a way determines a trajectory that encounters the defender’s hand at a certainpoint in space) and the rules of football (the handball determining a penalty kick).

If we think that grounding and causation are substantially unrelated concepts,the SEM analysis of this situation looks a bit artificial in that there is a formalcontinuity in the treatment of the chain of dependencies that leads from crossingthe ball into the penalty area to awarding a penalty kick yet a conceptual discon-tinuity involving some links of this chain. In other words, even if we depict thesituation as a seamless chain of dependent facts –which, by the way, perfectly fitsour intuitions–, from an ontological perspective, we nevertheless have two sepa-rate series of dependent facts, namely, (i) the crossing of the ball causing the ballhitting the defender’s hand, and (ii) the penalty kick being awarded in virtue ofthe handball being sanctioned.

If, instead, we are willing to accept that some sort of relation between causa-tion and grounding really holds, then no such issue arises. The dependence chainregains its conceptual unity through the relatedness of the two concepts, and thisalso explains why the SEM analysis of mixed chains perfectly fits our everydayunderstanding of the situation as a seamless sequence of facts. This is indeed akey point to be highlighted since, as we are going to see in a moment, dependencechains in general relativity are in fact mixed.

5 Dependence relations in general relativity: A pos-sible answer

Let’s step aside for a moment from the grounding/causation relatedness question,and let’s just try to apply the generalized SEM analysis to the case of geodesicmotions being dependent on geometrical facts in general relativity. As we aregoing to see, resorting to the SEM framework gives the issue a clear formal settingand, by doing so, brings to the fore the source of the confusion surrounding thedebate reviewed in sections 2 and 3.

Before doing this, however, let’s see how the generalized SEM framework isable to address the challenges that standard analyses of dependencies face in thecontext of general relativity. The following discussion comes with an importantcaveat, namely, that the application of the SEM framework to concrete cases fromgeneral relativity is still a mostly unexplored line of research: to my knowledge,the only paper that follows this approach is Vassallo and Hoefer (2019), where thequestion regarding the causal nature of frame-dragging effects in general relativityis analyzed in terms of generalized structural models. As a consequence, whatI am going to present here is the general conceptual strategy thanks to whichthe SEM framework deals with the issues related to dependence analysis. The

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concrete implementation of such a strategy would clearly depend on the particularcase studied.

The first issue regards the way we can evaluate counterfactual changes. Aswe have seen, given a solution 〈gµν,Tµν〉 of (1) representing the “actual” situation,there is in general no clear-cut criterion to single out the solution 〈gµν, Tµν〉 thatis “most similar” to 〈gµν,Tµν〉 and represents the implementation of the counter-factual change considered. To put it in the language of possible world semantics,there is in general no cogent way to single out the possible world nearest to theactual one because there is no spatiotemporal structure “shared” among possibleworlds that constitutes a standard against which counterfactual variation can beevaluated. By adopting the SEM framework, we solve the issue because we donot evaluate the truth values of counterfactuals directly against the solutions of(1), but against a given structural model, i.e. by computing the values of the cor-responding variables of the structural equations modeling a given situation. Usingfancy words, we can say that it is the particular structural model chosen that de-termines the modal horizon of a given situation, which means that counterfactualreasoning is structural model-dependent. For example, going back to the disap-pearing mass case of section 2, if the structural model we are working with isinformative enough and rightly captures the dependencies in the actual world, wejust have to set the material variables in the dependence chain to zero and checkto see what geometric facts obtain as a consequence in order to see which vacuumsolution of general relativity represents this new situation. If, on the other hand,by setting the material variables to zero we get a situation incompatible with thelaws of general relativity (e.g. geometrical facts not encoded in any vacuum solu-tion of (1)), this means that the structural model does not pass the test, and so it isnot accurate.

The problem of trans-world identification of spacetime points and regions isdealt with along the same lines. Under the SEM framework, any counterfactualchange in a spacetime region A in a solution of (1) just translates into a change ofvalue(s) of the variable(s) representing A in the structural equations of an appro-priate model. Again, if the structural model works well, this new situation will bedepicted by another solution of (1), which will thus represent the counterfactualsituation in which the very same A were to be changed. Note how this way ofdefining identity by stipulation is akin to what physicists do in concrete cases. Forexample, when dealing with a perturbation of a background metric, they map theperturbed spacetime against the unperturbed background in a way that associateseach point P of the latter to a point P′ of the former (see, e.g., Bruni et al., 1997,especially section 3).

Likewise, for counterfactuals involving small local changes in a solution of(1) that leave the rest of the universe untouched, all we have to do is to adopt

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an accurate, i.e. fine-grained enough, structural model that represents the actualdependence chain, “zoom in” the particular situation by looking at the variablesthat describe the local state of affairs, change their values as required, and thensee whether and how this change globally affects the dependence chain. Clearly,we cannot perform such an operation directly on a solution of (1). Nonetheless,we can use the structural model to “transition” from the starting solution of thefield equations (compatible with the starting values of the variables figuring in thestructural equations) to a new one which encloses the changed state of affairs. Thekey requirement, once again, is that the structural model be fully compatible withthe laws of general relativity, otherwise the changed state of affairs fixed by thenew values of the structural variables might not be encoded in any solution of (1).Also in this case, the SEM framework offers a clear strategy to bypass the problemwith model-wide interventions that Bartels’ proposal had.

The reader unsympathetic towards this generalized SEM framework mightpoint out that the general strategy sketched above does not eliminate the arbitrari-ness involved in the evaluation of counterfactuals, but just shifts it to a method-ological or pragmatic perspective: if a structural model works, then we can useit for counterfactual reasoning without being bothered too much by metaphysicalquestions about modality. This is a fair remark which, however, does not challengein any way the effectiveness of the SEM framework in treating counterfactual sit-uations in general relativity. The general strategy I have just presented might beas conceptually nasty and inelegant as you want (depending on your philosophicaltastes, of course), but still it provides a robust template for implementing struc-tural models that are able to cope with the usual troubles that standard analyses ofdependence face in general relativity.

Moving on, note how the SEM framework, in virtue of its loose and liberal in-terventionist attitude, does not require any notion of physical production in orderto define the wiggling of the variables, thus dodging any potential issue related tothe description of local physical processes in general relativity. If to this we addthat the SEM framework deals with facts rather than physical events, we under-stand why it handles cases of causation by absence without being committed to theactual physical production of absences. Hence, if –for the sake of the argument–we bite the bullet with Katzav’s challenge and claim that spacetime just causesthe absence of inexplicable changes in the trajectories of freely-falling bodies, wehave no problem to coherently render such a claim via a structural model thatresembles the plant watering case. Such a model would back up a perfectly fineexplanation of why freely-falling bodies do not move erratically, which does notmention at any point that spacetime literally produces the absence of erratic mo-tions.

Having argued that the SEM framework is a legitimate and promising ap-proach to analyzing dependencies in general relativity, let us apply it to the case of

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geodesic motions. Our model aims to account for the geometry of spacetime gµνdetermining the set of geodesic motions

{xµi (λ)

}i∈N

. In order to construct a generalmodel of this dependence that is compatible with general relativistic dynamics, allwe have to do is to follow the steps already sketched in section 2 when discussingBartels’ proposal.

If we set the exogenous variables to model certain facts about the metric tensoritself and we link their variation to the endogenous variables encoding facts aboutthe geodesic motions xµi (λ) via a certain number of functional relations symbol-ized by fi, in the end we get a set of structural equations that schematically looklike xµi (λ) ←= fi(gµν). To reiterate a point already made earlier, these equations neednot be literally (1), (2), and (3). These latter equations would have to be used ifwe were interested in providing a detailed picture of the geodesic motions over agiven spacetime, but here we want to construct a generic model that just capturesthe dependence of some facts on other facts in a theoretical context where the lawsof general relativity hold. Hence, the role of (1), (2), and (3) in this case is thatof a conceptual constraint: for a structural model to rightfully capture the map ofdependencies in a given situation, it has to be consistent with the laws of generalrelativity (hence, we cannot have situations where, say, by increasing spacetimecurvature in a region A the geodesic deviation decreases over A).

Given that we are interested in providing a general picture of how the analysiswork, we are not interested in the particular form that these functional relationscan take, the only constraint being that they capture, in the sense discussed above,the right counterfactual patterns encoded in general relativistic dynamics. So let’sjust assume that we get to a model with a chain of dependence of the form:

Facts about gµν Facts about{xµi (λ)

}Is this dependence causal or metaphysical? This is a clear case in which standardgrounding/causation demarcation criteria are ill-suited for the analysis. For exam-ple, the synchronicity/diachroneity criterion looks odd in a context where (i) thereis no “absolute” temporal ordering against which we should evaluate the relation-ship between facts, and (ii) some facts are about time itself. The same limitationsapply to the fundamentality/temporality criterion, especially given that the struc-tural equations do not carry any useful information to evaluate the dependencebased on this criterion. Fortunately, we can resort to the law-based demarcationcriterion proposed at the end of section 4. According to this criterion, all we haveto do is to look at the mediating principles that justify drawing the arrow in theabove graph. By doing this, we immediately realize that the main justificationcomes from the Einstein field equations, from which the equations of motion (2)follow. Given that the Einstein field equations are to be considered as laws ofnature, it looks like we have reached a verdict: the above dependence is causal

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in nature. However, this conclusion would be too quick because we need anotherpiece of information in order to justify such a dependence.

The problem is that the Einstein field equations do not single out a connection:many of them can go along with the same solution 〈gµν,Tµν〉. This is why two con-straints are placed ab initio on the connection, namely, (a) to be torsion-free, and(b) to be compatible with the metric tensor, i.e. to satisfy the relation ∇g = 0. Itis important to stress the fact that these constraints are not placed to preserve theinternal consistency of general relativity but, rather, they are theory-defining. Infact, relaxing or giving up on them would lead to entirely different but still con-sistent theories (see, e.g., Hehl et al., 1976). Hence, (a) and (b) can be consideredcontingent law-like conditions rather than mathematical constraints that hold bymetaphysical necessity. These two constraints are enough to single out a uniqueconnection because, according to the fundamental theorem of Riemannian geom-etry (whose proof can be found in any decent differential geometry textbook, suchas Lee, 2009), there is just one torsion-free connection compatible with a givenmetric tensor, that is, the Levi-Civita connection (whose components are in factgiven by (3)). We thus have to sharpen our model in order to bring this point tothe fore. To this extent, we add a further link in the chain:

Facts about gµν Facts about Γµνσ Facts about

{xµi (λ)

}We immediately see that the law-like condition (2) justifies the second link inthe chain.16 Instead, the justification for the first link comes from constraints (a)and (b) –which can be considered part of the laws of general relativity– and thefundamental theorem of Riemannian geometry, which is a mathematical result.We hence have a case of mixed dependence. This conclusion nicely explains theroots of the confusion surrounding the debate reviewed in sections 2 and 3: wesimply cannot force a standard reading that is purely causal or ground-like on thedependence chain depicted above.

A possible objection to the analysis carried out so far is that it is totally mis-guided, since it presupposes a one-way determination relation from geometricalto material facts, where in fact the dependence relation that best depicts the Ein-stein field equations is mutual: after all, this is exactly what the Misner, Thorne,and Wheeler motto is all about! To this I reply that, although it is true that thereis no arrow on top of the equal sign in (1), still these equations do not depict amutual dependence tout-court. All the Einstein field equations say is that there isno “supremacy” of material degrees of freedom over geometrical ones, and vicev-

16It is worth mentioning that the derivation of (2) from (1) through the condition that the co-variant divergence of the stress-energy tensor vanishes works only if the Levi-Civita connection isused (thanks to an anonymous reviewer for suggesting this point to me).

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ersa. Returning to the dependence chain depicted above, for sure we can render itmore accurate by adding a further link upstream, thus having:

Facts about Tµν Facts about gµν Facts about Γµνσ Facts about

{xµi (λ)

}Now, the first link in this chain represents how material facts determine geomet-rical facts. However, there is no mutuality at all involved in this chain, since thematerial facts upstream are entirely different from the material facts downstream.For example, the first node may describe the overall material distribution of a bi-nary system (e.g. on a Cauchy surface), while the last may describe the orbits thatthe system follows.

The point becomes even stronger if we enlarge the upstream part of the chain,remembering the functional dependence of the stress-energy tensor:

Facts about Φ

Facts about Tµν Facts2 about gµν Facts about Γµνσ Facts about

{xµi (λ)

}

Facts1 about gµν

Note that the lower first node does not coincide with the third and, likewise, theupper first node does not overlap with the second. Take, for example, the case ofthe Schwarzschild solution to (1) in the interior of a star (see Wald, 1984, section6.2, for a derivation of this solution). In this case “Facts1 about gµν” include thatthe geometry is static and spherically symmetric, while “Facts2 about gµν” involvea more detailed characterization of the geometry, including its dependence onthe total mass of the body. Similarly, “Facts about Φ” include that matter is afluid with such and such density and pressure, while “Facts about Tµν” involvea detailed characterization of the stress-energy tensor, including the fact that the4-velocity of the fluid is such and such as a consequence of spacetime being static.

Coming back to a general discussion of the above chain, note that neither ofthe two first nodes is dependent on the other, which represents the fact that, ac-cording to general relativity, there is no material fact that is more fundamentalthan all geometric facts, and viceversa. In this sense, the analysis carried out inthis paper is fully faithful to Misner, Thorne, and Wheeler motto (whose represen-tation is in fact encoded in the above chain), and to the dependencies encompassedin (1). Moreover, note how, by expanding the dependence chain, its mixed char-acter is reinforced. Indeed, now we have that the first (double) link is justifiedby (mathematical) functional dependence, while the second invokes the Einsteinfield equations plus boundary conditions.

In the end, what are the morals to be drawn from the above analysis? If wetake seriously the generalized SEM framework, including the law-based demar-

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cation criterion, then we have to conclude that the dependence relation under-pinning general relativistic dynamics is definitely neither pure causation nor puregrounding.17 This, by itself, is not a huge achievement: already at the end ofsection 3 we reached the conclusion that this relation is too “physics-friendly” tobe just grounding yet not “physically active” enough to be just causation. How-ever, the generalized SEM framework supplies us with a new perspective on theissue. Indeed, while before we assumed that grounding and causation were totallyunrelated concepts, and this assumption made the conceptual tangle impossibleto loosen, now the SEM framework –being a unified analytical framework forgrounding and causation– suggests to us that a way out of the impasse is in fact toaccept that the two concepts are related in a substantial metaphysical sense, andgeneral relativity is a theory that makes this relatedness manifest. This, of course,begs the question: in what way are grounding and causation related?

According to Wilson (2017), they are just one and the same thing. Otherwisesaid, grounding is just a “metaphysical” way of causing. Hence, there is just aunique fundamental dependence relation –which is a conceptual primitive–, whichmay exhibit different “flavors”. Wilson calls this fundamental relation “causation”and distinguishes the metaphysical flavor (which we would call “grounding”)from the nomological one (which we would call “standard causation”). However,this is just a matter of terminology. The gist of his thesis is that, where we previ-ously saw two conceptually distinct dependence relations, now there is in fact justone dependence relation at work. The benefits of adopting this view are ratherobvious: we get a parsimonious ideology and a unified explanatory frameworkwith a relatively small effort.

Under this view, there are no mixed chains of facts, since there is only one re-lation involved. Indeed, the distinction between laws of nature and metaphysicalprinciples is useful only insofar as it explains why, up to now, metaphysicians hadthe impression that grounding and causation were two conceptually distinct rela-tions: simply, they have just focused on a distinction that cuts no metaphysical iceas far as the relation of determination is involved. Such a “metaphysical illusion”of there being two distinct dependence relations stems from the fact that meta-physicians usually focus on clear-cut, well-behaved everyday situations. When,however, fundamental physics comes into play, common intuitions are thrown outof the window, and the metaphysical unity of dependence relations becomes man-ifest, as the case of general relativity shows.

17Note that a similar conclusion would have been achieved even if we had adopted an alternativeapproach to the construction of general relativity, such as that of Ehlers et al. (1972). Accordingto this approach, the direction of the dependence depicted on page 26 would have to be reversed.However, the analysis of the nature of the dependence involved would proceed similarly to the onepresently carried out (the same mediating principles would have been invoked, only in a differentorder).

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Wilson’s thesis is rather strong and surely controversial (see, e.g., Schaffer,2016, section 4.5, for a critique of the grounding/causation unification thesis).However, we do not need to go that far to clarify the issue at stake. For exam-ple, we could maintain that, while grounding and causation are distinct concepts,still they are related in that they represent two determinates of a common deter-minable.18 Under this view, the law-based demarcation criterion determines ametaphysical scale of nomic dependence, causation being at one end (full nomicdependence) and grounding being at the other end (full non-nomic dependence).Therefore, the dependence relation underpinning (1) falls somewhere in the mid-dle of this scale, being some sort of metaphysical hybrid between causation andgrounding. This explains why other analytical frameworks, which do not envisagethis nomic dependence scale, are unfit to clarify the matter.

This is clearly not the place for adjudicating the dispute between dependencemonists and pluralists. More modestly, the point that I want to drive home isthat, by adopting a generalized SEM framework, it is possible (i) to elaborate astrategy to overcome the common conceptual troubles that standard analyses ofdependence have in general relativity and (ii) to come up with some clear an-swers regarding the metaphysical status of the dependence relation encoded inthe Einstein field equations. Of course, future work will have to show how theframework overcomes said conceptual troubles in concrete cases. Moreover, bothmonists and pluralists still have to deepen their own characterization of the dis-cussed dependence relation –perhaps by looking at other fundamental physicaltheories. However, even at this earlier stage, I hope to have shown that the appli-cation of the generalized SEM framework to general relativity is a promising andexciting line of research, which intertwines physics with metaphysics.

Acknowledgements:

Many thanks to Carl Hoefer, Vassilis Livanios, Al Wilson, and two anonymousreferees for their comments on earlier drafts of this paper. Ça va sans dire, I amsolely responsible for any remaining frown-inducing material. Also, I acknowl-edge financial support from the Spanish Ministry of Science, Innovation and Uni-versities, fellowship IJCI-2015-23321.

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